Mathematics - Indefinite Integration Question with Solution | TestHub

$$I=\int \frac{dx}{{\left[(x-1{)}^3(x+2{)}^5\right]}^{1/4}}\Rightarrow I=\int \frac{dx}{(x+2{)}^2{\left[\frac{(x-1)}{(x+2)}\right]}^{3/4}}$$

Let $t=\left(\frac{x-1}{x+2}\right)$.

$$\begin{array}{rl}dt& =\frac{(x+2)-(x-1)}{(x+2{)}^2}dx\\ dt& =\frac{3}{(x+2{)}^2}dx\\ \Rightarrow \frac{dt}{3}& =\frac{dx}{(x+2{)}^2}\\ I& =\int \frac{dt}{3t^{3/4}}\\ I& =\frac{1}{3}\int t^{-3/4}dt\\ I& =\frac{1}{3}\frac{t^{1/4}}{1/4}+C\\ I& =\frac{4}{3}{\left(\frac{x-1}{x+2}\right)}^{1/4}+C\end{array}$$

Comparing this with the given form, we have $p=4$ and $q=3$, so $p+q=7$.